and Probability Algorithms Leonard J. Schulman Notes for lecture 12, Feb 24, 2003. Sampling graph colorings.
Caltech CS150, Winter 2003 Scribe: Wonjin Jang
Let be the number of vertices and be the max degree in G. We wish to sample -colorings uniformly. For the decision problem, the following is known - If , it's a computationally hard problem. If , there always exists a solution. If , there is a polynomial time algorithm to solve the decision problem. (There are just a few special graphs which cannot be -colored, and we can recognise them.) For the sampling problem (sample approximately uniformly at random among all colorings), the status is that if , there is a poly-time sampling method. As an aside, it is known that the number of colorings of a graph can be evaluated exactly by evaluating a certain polynomial called the chromatic polynomial of the graph. The above stated facts imply that this polynomial can be (computationally feasibly) closely approximated for , but not for unless P=NP. The cases in between are open.
We'll show poly-time approximately uniform sampling in the case that (proving the result for is outside the scope of this class) and prove it by a coupling on q-colorings of G. The Markov Chain is as follows: Pick uniformly. Recolor to unless some neighbour of is already colored with c. This Markov chain is 1. Non-periodic, 2. Connected. These conditions imply ergodicity of the chain.
Call the step "good" if the recoloring can be done in both chains, and if and had not agreed at prior to the recoloring. Call the step "bad" if the recoloring can be done in just one chain, but and had agreed at prior to the recoloring. (Note that a step may be neither good nor bad.) 1
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The coupling process is straightforward: choose a vertex recolor to in each coloring if it is possible.
and a color
uniformly and randomly, and
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Coupling: Let and be two colorings of the graph, represented graphically in the above figure. Let be the set of vertices in which agrees in color with . Define the distance between two colorings as
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1 Random sampling of graph colorings with
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if the last step was "good"
if the last step was "bad" otherwise
The total number of moves available to the algorithm in each step is The number of "good" moves is and each of these neighbors has one color in recolored.
.
coupling time
This method can extended be to
with a bit more work.
is harder.
2 Approximatly counting the number of colorings
Now that we know how to sample approximately uniformly from the colorings of a graph, subject to , let's see how to count those colorings. (This doesn't follow automatically because the coloring problem isn't self-reducible.)
We will estimate
by estimating each ratio
, and multiplying out.
We will estimate each of the ratios by sampling nearly uniformly from and checking whether the coloring is in . This will give us a reliable multiplicative estimate of the ratio because of: 2
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We are given a graph with edges and maximum degree , and a parameter edges arbitrarily . Let be the subgraph of with edges . Let Then we have the following relations.
=
. Order the -colorings of .
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(as above) because a bad move is an attempt to recolor a neighbor The number of "bad" moves is of a vertex , with one of the colors or .
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, since there are at most neighbours for each , and another (possibly equal) in , with which
, cannot be
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Run...